Hi, I'm actually reading a poker theory book and I want to test a toy game.

If I am given one of 3 cards : A, K, Q equally often, and I bet all my A, and I bet 1/3 of my Q, and I check all my K.

Then, I really don't see how is my range after I check.

I understand I have no more A, I have 2/3 of the Q I had, and all my K.

But which proportion of the range is K, and which is a Q now, I don't see the formula to find the answer.

I guess I could simplify things and say like 33.33 * 2/3 + 33.33, and then my range is 22.22/66.66 Q, so 1/3 Q and 2/3 K, but it's kind of hard for my brain to process. There should be to just use a formula to find the answer ?

Thank you

You have all the information you need to determine the "weighted" check range.

You can reason it out in several ways:
(1) use common sense
(2) use the "probability weightings" of K vs Q checks
(3) use Bayes Theorem
(4) write down the entire decision tree.

Of course, there is a similar "weighted" bet range. Maybe that is easier to think about.

Actually I realize this is more easy than I thought.

I should just see the ratio of Q to K, and divide by the total of Q+K.

So since I have 2/3 of my Q, and I have 3/3 of my K, I should do 2/(2+3) and I have 40% of Q.

However, the next step I am confused is if in the toy game we are both allowed to bet half pot (no raise allowed), and I check, then when my opponent bets, I know I must defend 2/3 of the time otherwise he can exploit me.

But, I am confused, since my range is now 6/10 K and 4/10 Q, I could not even defend enough with my K.

However, the author seems to suggest that I should defend 1/3 of my K, because A takes the burden to defend the rest. (because I could check my A).

so if I would check my whole range, when I face a bet from a Q actually, my range would be 3/6 A, and 3/6 K, so I would defend 1/3 of my K to defend 4/6 of the time because euh.. 1/3 of the K is 1/6 of the range + 3/6 of A, I can now defend 2/3 of the time with only 1/3 of the K.

BUT, If I would bet all my A and 1/3 of my Q, then if I would still defend 1/3 of my K only, it seems like I don't defend enough. It seems to me that if I still only defend 1/3 of my K, and I check and face a bet from the Q, I will fold 2/3 of the time because when my opponent have a Q my real range would be K.

But then my opponent, who should bet all A, and 1/3 of his Q, should now bet all of his Q ? I am very confused because the author seems to suggest that whatever balanced set of strategy I would use as out of position player has same EV. But I am not sure if he consider that our defending range is not the same after we check.

Hi, could you please delete this post. I think I found the solution.